Math 117 Chapter 2 Systems of Linear Equations and Matrices


 Corey Wood
 1 years ago
 Views:
Transcription
1 Math 117 Chapter 2 Systems of Linear Equations and Matrices Flathead Valley Community College Page 1 of 28
2 1. Systems of Linear Equations A linear equation in n unknowns is defined by a 1 x 1 + a 2 x a n x n = k where a 1, a 2,..., a n and k are real numbers and x 1, x 2,..., x n are the variables. An equation is nonlinear if any of the variables are raised to a power, multiplied together or involve logarithms, exponentials or trigonometric functions. We do not deal with nonlinear equations in this course. A system of linear equations is a set of one or more linear equations. In both the Supply/Demand and the BreakEven Analysis problems in Chapter 1 you derived two linear equations with the goal of finding where the two equations were equal to each other. In each case a system of linear equations was solved. A solution to the system is the set of point or points that satisfy all of the equations Possible Solutions to a System of Linear Equations Given any system of linear equations there are three possible outcomes when trying to solve the system. 1. One Solution: There is one and only one set of points (x 1, x 2,..., x n ) that satisfy all of the equations. This system is said to have a unique solution. With two equations and two unknowns the solution is the point where the graphs of the equations intersect. Page 2 of 28
3 2. No Solution: There are no points that satisfy all of the equations. With two equations and two unknowns the graphs of the equations are parallel with different y intercepts. In this case there is no point that satisfies both equations. 3. Infinitely Many Solutions: There are infinitely many points that satisfy every equation in the system. With two equations and two unknowns graphs of the equations are identical. Warning: Infinitely many solutions does not mean any set of points will satisfy the system. What this does mean is that any point that satisfies one equation will satisfy all of the equations. Now for just a little more vocabulary. Any system that has a solution, one or more is called consistent. If no solution exists, the system is called inconsistent. A system that has infinitely many solutions is called dependent. As will be shown later, the solutions to a dependent system will be determined by choosing any value for one or more of the variables which in turn will determine the values of the remaining variables. If the is one or no solutions, the system is called independent. The following table summarized the vocabulary with the possible outcomes of systems of linear equations. One Solution Infinitely Many Solutions No Solution Consistent Consistent Inconsistent Independent Dependent Independent Page 3 of 28
4 1.2. Equivalent Equation Operations One method used to solve systems of linear equations is to transform the original system into a different system that has the exact same solution(s). The transformations must make the system simpler without changing the answers. If the new system has the same solution, the system is called an equivalent system. There are three transformations that result in an equivalent systems. 1. Equation Exchange: The order of any two equations can be switched. 2. Multiply an Equation by a Constant: Both sides of an equation can be multiplied by a nonzero constant. This transformation is usually performed when there is a common factor in every term of the equation or one wishes to remove fractions from an equation. 3. Add a Multiple of One Equation to Another Equation: This transformation will be the most useful solving systems. It is very important to really understand this transformation as one equation will be changed drastically with this action. The new equation replaces the equation added to, not the equation that was multiplied by the constant. For example if one wishes to multiply the first equation by 5 and to the third equation, it is the third equation that is changed. The first equation remains exactly the same. Page 4 of 28
5 1.3. Echelon Method The first method in solving systems is called the Echelon Method. The goal of the Echelon Method is the use the equivalent equation operations to rewrite the system in to what is called a triangular form. For three equations with three unknowns triangular form looks like x + ay + bz = c y + dz = e z = f. (E1) (E2) (E3) Once triangular form has been achieved it is easy to solve the system using backsubstitution. Backsubstitution starts at the bottom of the system works back up the system. In the example above the value for z is given in equation 3. Substitute the value for z in equation 3 back into equation 2 and solve for y. The final step is to substitute the values for z and y back into equation 1 and solve for x. To find the triangular form of a system start at the upper left corner, usually with the variable x, and use equation operations to eliminate x in all of equations below. Move down one equation and to the right and repeat the elimination process. This is a very mechanical method that once mastered will allow one to solve a system with many equations and many variables. Page 5 of 28
6 Inconsistent System Triangular form is very useful to determine if the system is consistent or not. If the system is inconsistent the last equation will lead to an impossible equation such as 0 = 5. At this point stop, and state the system has no solution. Dependent System If the final equation ends up with 0 = 0, then the system is probably dependent. In the three equation three unknown case this will look like x + ay + bz = c y + dz = e (E1) (E2) 0 = 0. (E3) Now both x and y depend on the the variable z. The z variable is now called a parameter or the free variable. Choosing any value for z and substituting back into equation 1 and 2 will result in one solution to the system. Since z can be any number, there are now infinitely many solutions to the system. Page 6 of 28
7 2. Matrices and Systems of Linear Equations After practicing solving systems using the Echelon Method one will quickly find it a bit cumbersome to keep track of all the variables in the equation operations. Matrices can be used to rewrite the system in a more manageable format. A matrix is nothing more than an array of numbers. Each number in the matrix is called an element or entry. The element of a matrix will be referenced by their location in the matrix. To be consistent the entries will be identified first by the row and then the column location in the array. Before rewriting the system it is important that all of the variables are aligned in the same vertical location. Now by taking only the coefficients of the variables the system can be written as an augmented matrix. For three equations and three unknowns the x coefficients will go in the first column, the y coefficients in the second column, the z coefficients in the third column and the last column will be all of the constants to the right of the equal sign. It is very important to keep track of what every column represents. For example the system has an augmented matrix x + 3y 6z = 7 2x y + 2z = 0 x + y + 2z = Page 7 of 28
8 Notice that the row of the augmented matrix gives the coefficients of the corresponding equation. Now the equivalent operations for the augmented matrix are almost identical to the equivalent equation operations Equivalent Row Operations 1. Row Exchange: Any two rows can be switched. 2. Multiply an Row by a Constant: A row can be multiplied by a nonzero constant. 3. Add a Multiple of One Row to Another Row: Again this operation will be the most useful solving the matrix problem. As before it is very important to understand the new row replaces the row added to, but the row that was multiplied by the constant stays the same. For example if one wishes to multiply the first row by 5 and to the third row, it is the third row that is changed. The first row remains exactly the same. Page 8 of Gauss Method The Gauss Method is the same method as the Echelon Method but uses matrices and equivalent row operations instead of equations and equation operations. In fact, with an augmented matrix the Gauss Method is often called the Echelon
9 Method. The goal is to rewrite the augmented matrix to the triangular form A matrix of this form is said to be in rowechelon form and s will be real numbers. To finish solving the system rewrite the triangular matrix as a system of equations and use back substitution to solve. In rowechelon form the first nonzero entry in each row is called a pivot. While it is desirable to have the pivots all ones, this is not necessary. Often it is easier to leave the pivots as they are until triangular form has been established. Then at the end use multiplication by a constant to make the pivot a one. In rowechelon form a system with no solution will look similar to Here the last row of the matrix is written 0 = 5, an impossibility. In rowechelon form a dependent system will result in a last row of zeros Notice there is no coefficient for the z variable. In this case z is referred to as a free variable. Page 9 of 28
10 When solving a system using the augmented matrix it is very important to start in the upper left entry, establish a pivot in the matrix and use row operations to get zeros for all of the other entries below the pivot. Next move down and to the right to the first nonzero entry for the next pivot and get zeros below that pivot. Repeat this process until the desired triangular form is produced. Warning: Deviating from the process of working from upper left to lower right will not lead to the desired triangular matrix GaussJordan Method It is possible to skip the algebraic back substitution entirely and still arrive at the correct solution to the system. This is call the GaussJordan Method or reduced rowechelon form. After row operations have been used to achieve rowechelon form, start form the lower right and use row operations to get zeros above the pivot. Move up and to the left to the next pivot and repeat until the matrix is of the form Now it is just a matter of writing down the solution. When there is no solution to the system, there is no need to continue from rowechelon form. When there is no solution, there will still be no solution after more row operations. Reduced rowechelon form is very useful when there are infinitely many solutions. The Page 10 of 28
11 columns with a pivot are the dependent variables and those without pivots are the independent variables Solving Systems with the Calculator With a little practice solving simple systems of two or three equations with two or three unknowns is not that difficult. For these simpler systems you will be expected to show your work to solve the systems using row operations on the augmented matrix. For larger systems and to check solutions to simpler systems the calculator can be a valuable and time saving asset. 1. Enter the Augmented Matrix into the Calculator: Once the system has been written as an augmented matrix the matrix is entered into the calculator. Press 2ND then MATRIX to enter the matrix environment. Move over to EDIT and down to any of the matrices your wish, eg. 1:[A], and press ENTER. First set the size of the matrix, called the dimension of the matrix. The dimension of a matrix is always the number of rows number of columns. Now enter the entries for the augmented matrix. After all of the entries have been entered, go back to the main screen using 2ND QUIT. To check the augmented matrix enter the matrix environment, 2ND MATRIX, and select 1: [A] 3 4. For example the system x + 3y 6z = 7 2x y + 2z = 0 x + y + 2z = 1 Page 11 of 28
12 appears in the matrix environment as 2. RowEchelon Form: To express the augmented matrix in rowechelon form, ref, press 2ND then MATRIX to enter the matrix environment. Arrow over to MATH, arrow up to A:ref( and press ENTER. Again enter the matrix environment, 2ND MATRIX, and select the matrix where the augmented matrix was entered, 1:[A]. Finally close the parentheses, ) and press ENTER. The resulting matrix will be in rowechelon form. Note: Rowechelon form is not a unique operation. Do not be surprised if the calculator s results do not match your hand calculations. For the previous system the rowechelon form is Page 12 of Reduced RowEchelon Form: The procedure to express the augmented matrix in reduced rowechelon form, rref, is very similar to rowechelon
13 form. Press 2ND then MATRIX to enter the matrix environment. Arrow over to MATH, arrow up to B:rref( and press ENTER. Again enter the matrix environment, 2ND MATRIX, and select the matrix where the augmented matrix was entered, 1:[A]. Finally close the parentheses, ) and press ENTER. The resulting matrix will be in reduced rowechelon form. Reduced rowechelon form is a unique operation. Now your handcalculations should match the calculator. The reduced rowechelon form for our example is The solution to the system is x = 1, y = 0, z = 1. As before, we are not going to use spreadsheets to solve systems, method 3 in the textbook. You will, however, be responsible for hand calculations as well as calculator solutions to solve systems of linear equations. Page 13 of 28
14 3. Matrix Addition and Subtraction In the previous section matrices were used to solve systems of linear equations. The applications that use matrices is far greater than just systems of linear equations. In chapter 1 the calculator was used to find the linear function to best fit data. This is actually a very involved matrix application. At the conclusion of this chapter we will investigate the use of matrices in the economic application of InputOutput models. Before we get to the applications the arithmetic of matrices must be discussed. The main components of any arithmetic are addition, additive identity, additive inverse (subtraction), multiplication (two types), multiplicative identity and multiplicative inverse (not division). Once the operations and identities are understood it is important to establish the appropriate properties of the arithmetic that hold. These properties include the associative, commutative and distributive laws. While many of the properties of real numbers will be the same for matrices, there are a couple of surprised along the way Matrix Vocabulary and Definitions When entering matrices into the calculator the size or dimension of the matrix had to be known. A matrix A with m rows and n columns is defined to be an m n matrix, written dim(a) = m n. A square matrix has the same number of rows and columns, thus has dimension n n. A row matrix or row vector has dimension 1 n. A column matrix or column vector has dimension m 1. In general capital letters will be used to represent a matrix while lower case letters Page 14 of 28
15 will represent elements of the matrix. Subscripts will be used to indicate the location of the element of a matrix. Remember that for matrices it is row first, then column. For example, given the matrix A = The element in the second row and third column is 2 and will be written a 23 = 2. In general the element in the i th row and j th column is written a ij. For a matrix with dimension m n it is also often neater to express the matrix A as A = [a ij ] for 1 i m and 1 j n instead of writing out the entire matrix. This notation will be used to define the arithmetic operations on matrices. For example, Matrix Equality Two matrices are equal if they have the same size and if each pair of corresponding elements is equal. That is, A = B if and only if dim(a) = dim(b) and a ij = b ij for all i and j Matrix Addition Now that the basic definitions and notation are understood it is time to look at the arithmetic of matrices. Once matrix addition has been defined an identity matrix and additive inverse follow directly. Page 15 of 28
16 To verify these definitions it will be useful to enter the following matrices into your calculator: [ ] [ ] [ ] A =, B =, C =, Adding Matrices D = [ 2 1 ], E = [ 3 5 ], F = [ ] [ ] 3 1, G =. 2 2 The sum of two m n matrices X and Y is the m n matrix X +Y in which each element is the sum of the corresponding elements of X and Y. Or equivalently, given X = [x ij ] and Y = [y ij ], then X + Y = [x ij + y ij ]. Try using your calculator to add matrices with the same dimension, A + B, as well as matrices with different dimensions, A + D. Identity Matrix for Addition An additive identity is a matrix, 0 such that A + 0 = 0 + A = A. The identity for matrix addition is called the zero matrix. The zero matrix adjusts its dimension to match that of the matrix A. The zero matrix is an m n matrix with 0 for every entry. Page 16 of 28
17 Additive Inverse Given an m n matrix A, the additive inverse or negative of A is the m n matrix A such that A + ( A) = A + A = 0. Given A = [a ij ] the additive inverse is the defined by A = [ a ij ] where each element of A is multiplied by 1. Subtracting Matrices Now that the negative of a matrix has been defined subtraction of matrices follows exactly as with real numbers. For two m n matrices X and Y, the difference X Y is the m n matrix defined by X Y = X + ( Y ) Equivalently, given X = [x ij ] and Y = [y ij ] the difference X Y is defined by X Y = [x ij y ij ]. Page 17 of 28
18 4. Up to this point matrices have closely followed the properties and definitions of real numbers. Matrix multiplication will be the first deviation from the real numbers. To begin with there are two different matrix multiplications, scalar multiplication and matrix multiplication Scalar Multiplication A scalar is defined to be any real number usually represented by lower case k, l, m or n. Scalar multiplication most closely represents the repeated addition of real numbers. For real numbers is the sum of the number 3 five times or 5 3. For matrices the definitions is similar, A + A + A + A + A is defined as 5 A or just 5A. More formally, Product of a Matrix and a Scalar The product of a scalar k and a matrix X is the matrix kx, each of whose elements is k times the corresponding element of X, kx = [kx ij ] The ultimate goal of this section is to determine the requirements and methods of multiplying two matrices. Before jumping to the general case consider only the product of a row matrix and a column matrix. Using the matrices entered into Page 18 of 28
19 your calculator earlier, try multiplying D F. Now try D G. In both cases a 1 2 matrix multiplies a 2 1 matrix and the product is a 1 1 matrix. Now notice D F = [ 2 1 ] [ ] 3 = 2( 3) + 1(2) = = 4 2 and D G = [ 2 1 ] [ ] 1 = 2(1) + 1( 2) = = 0 2 Row Matrix times a Column Matrix In order to multiply a row matrix times a column matrix the number of elements in the row must be the same as the number of elements in the column. The resulting product will be a 1 1 matrix whose sole element is the sum of the products of the corresponding elements in the row and column. Now is probably a good time to point out that order is very important here. The definition above is only for (row matrix) (column matrix). Use your calculator to try D F. This leads to our first big difference between the real numbers and matrices. In general, matrix multiplication is not commutative, A B B A. From this point forward make certain the row matrix is to the left and the column matrix is to the right. [ ] 3 1 Next notice that the columns of matrix B = are identical to F = [ ] [ ] and G =. Any guesses to the product D B? Try it. Notice a couple 2 2 Page 19 of 28
20 of things: 1. dim(d) = 1 2, dim(b) = 2 2 and dim(db) = 1 2. In general, the number of columns of the first matrix must equal the number of rows in the second matrix in order to multiply row by column. 2. The entries in DB are the same as DF and DG, respectively. DB = D [ F G ] = [ D F D G ] In other words, to find the entry in the first row and first column of the product of DB, multiply the first row of D by the first column of B. To find the entry in the first row and second column of the product of DB, multiply the first row of D by the second column of B From these examples the definition of matrix multiplication arises. Product of Two Matrices Let A be an m n matrix and B be an n k matrix. To find the element in the i th row and j th column of the product matrix AB, multiply the i th row of A by the j th column of B. The product matrix AB is an m k matrix. To reiterate there are two very important properties of matrix multiplication: 1. AB is defined if and only if the number of columns of A is the same as the number of rows of B. If dim(a) = m n and dim(b) = n k, then dim(ab) = m k. 2. In general matrix multiplication is not commutative, AB BA. Page 20 of 28
21 5. In matrix arithmetic there is no division, only a multiplicative inverse. Before an inverse matrix can be defined one must define a multiplicative identity for matrices Identity Matrix For real numbers the multiplicative identity is the number 1. For any real number a, 1 a = a 1 = a. Taking a similar requirement for matrices an identity matrix,i, must satisfy A I = I A = A. In this definition the commutative property not only holds, but is required. In order for the multiplication to be defined both the identity I and the matrix A must be square. Identity Matrix The identity matrix is a unique square matrix defined as { e ij = 1 for i = j, I = [e ij ] such that e ij = 0 for i j. Do not be alarmed by the notation, the identity is a square matrix with all zeros except along the main diagonal. Below are the 2 2, 3 3 and 4 4 identity matrices. Page 21 of 28
22 I 2 = [ ] 1 0, I = , I 4 = Once the identity matrix is defined it is relatively easy to define an inverse for a matrix A. Multiplicative Inverse Matrix Given a square matrix A the inverse of matrix A, denoted A 1 is a matrix such that A A 1 = A 1 A = I. Recall the real number 0 has no multiplicative inverse, there is no number a such that a 0 = 1. Just like the real numbers not every square matrix A has an inverse. Actually finding a multiplicative inverse of A is much more work. Page 22 of Finding While defining A 1 is relatively straight forward, finding a multiplicative inverse for A is much more work. The formula for finding the inverse of a 2 2 matrix
23 will be given. For all higher dimensions the calculator will be used. Inverse of a 2 2 Matrix [ ] a b Given A =, the inverse of A is c d A 1 = 1 ad bc [ d ] b c a Notice the scalar in front of A 1, 1/(ad bc). If ad bc = 0 a division by zero error is encountered. In this case the matrix A does not have an inverse. For the 2 2 case it is often faster to check to see if ad bc = 0 and use the formula above than to use your calculator to find the inverse. by Calculator For square matrices with dimension larger than 2 2 we will use the textbooks method 2 only. Find an inverse matrix is actually a very computationally challenging task that is executed easily with your calculator. 1. First enter the matrix A into your calculator and return to the computation screen. 2. Select the matrix A from the matrix environment, 2ND, MATRIX, 1:[A]. 3. Press the x 1 button (This is actually the same button you have been using to enter the matrix environment.) and press ENTER. Page 23 of 28
24 4. More than likely the resulting matrix has a lot of decimals. It is usually possible to express the inverse matrix with fractional entries. Press MATH, 1: Frac and then ENTER Solving Systems using To end a very long chapter let us return to the beginning, solving systems of linear equations. Earlier we solved the system x + 3y 6z = 7 2x y + 2z = 0 x + y + 2z = 1 using an augmented matrix and the rref command. Consider the system as two equal column matrices, x + 3y 6z 7 2x y + 2z = 0. x + y + 2z 1 The trick here is to consider only the left side of the equation and realize the column matrix is really the product of two matrices, one known and one unknown. x + 3y 6z x 2x y + 2z = y. x + y + 2z z Page 24 of 28
25 The known matrix is called the coefficient matrix. This is nothing more than the augmented matrix without the column of constants. Let the coefficient matrix be called A and the unknown matrix x x A = 2 1 2, x = y z Do not confuse the column matrix x with the variable x in the system. Finally let the column of constants be labeled b. 7 b = 0. 1 Now the system is of the form Ax = b. Solving Ax = b Using Matrix Inverses To solve a system of equations Ax = b, where A is the matrix of coefficients, x is the matrix of variables, and b is the matrix of constants 1. Find A Multiply both sides of the equation on the left by A 1 to get x = A 1 b. Page 25 of 28
26 6. The final application of this chapter brings together all of the matrix arithmetic learned into a very useful economic modeling technique. In this model it is assumed there are n commodities produced. Each commodity uses some or all of other commodities in its production. An inputoutput matrix, or technological matrix, is used to represent what commodities are used to produce one unit of another commodity. Each column represents the commodity being produced and each row represents the amount of the commodity used in production of the commodity in the column. For example the inputoutput matrix A may represent the production of Agriculture, Manufacturing and Transportation. Agriculture Manufacturing Transportation 1 1 Agriculture 0 A = Manufacturing Transportation The first column shows that 1/2 unit of manufacturing and 1/4 unit of transportation will be used to create one unit of agriculture. A production matrix is a column matrix representing the total number of units of the commodity produced. Keep in mind that some of the units produced are used to produce other commodities. Thus the production matrix is a gross production, not the total units available. To find the number of units available, the number of units used in production must be subtracted from the production matrix. Multiplying the inputoutput matrix by the production matrix,, AX, will Page 26 of 28
27 yield the number of units used in production. Finally the total units available for sale is given by X AX. In practice the production matrix is not given, but must be found. In general an economy will be able to use a given quantity of a commodity. The quantity the economy desires to use is called a demand matrix, D. The demand matrix is also given as a column matrix. Now the producers of the commodities want to have enough units available after production to satisfy the demand. In matrix language the amounts to solving the matrix equation X AX = D. A little matrix algebra allows one to solve for X X AX = D (I A)X = D X = (I A) 1 D There is yet another method that may be used to find the production matrix, X. Once the matrix equation is in the form (I A)X = D it may be written as an augmented matrix [(I A)D] and use reduced rowechelon form to produce the desired result. Personally the second method is more appealing and less computationally expensive. The problem above is called an open model. In the open model it is the goal to have some surplus of the commodity to meet the demand of the public. In a closed inputoutput model there is no surplus, X = AX. A closed model may be desirable in a commune or a community that is self sufficient. Now the matrix problem amounts to solving (I A)X = 0. In this case the augmented matrix must be used. The augmented matrix in the closed case will have a column of Page 27 of 28
28 zeros that will not change during row operations. Thus it is only necessary to find rref([i A]. Two very important things to remember. 1. Don t for get the column of zeros in the last column of rref([i A]). Include the zeros before rewriting the solution equations. 2. There will always be infinitely many solutions. That is there will be a free variable in rref([i A]). In most cases choose the smallest value for the free variable that leaves whole number solutions. Page 28 of 28
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationLinear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More information4.34.4 Systems of Equations
4.34.4 Systems of Equations A linear equation in 2 variables is an equation of the form ax + by = c. A linear equation in 3 variables is an equation of the form ax + by + cz = d. To solve a system of
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationArithmetic and Algebra of Matrices
Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational
More informationSECTION 8.3: THE INVERSE OF A SQUARE MATRIX
(Section 8.3: The Inverse of a Square Matrix) 8.47 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX PART A: (REVIEW) THE INVERSE OF A REAL NUMBER If a is a nonzero real number, then aa 1 = a 1 a = 1. a 1, or
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More information5.5. Solving linear systems by the elimination method
55 Solving linear systems by the elimination method Equivalent systems The major technique of solving systems of equations is changing the original problem into another one which is of an easier to solve
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationRow Echelon Form and Reduced Row Echelon Form
These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for inclass presentation
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationSolving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationMatrix Algebra and Applications
Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2  Matrices and Matrix Algebra Reading 1 Chapters
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationLecture Notes 2: Matrices as Systems of Linear Equations
2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably
More information1.2 Solving a System of Linear Equations
1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems  Basic De nitions As noticed above, the general form of a linear system of m equations in n variables
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationAbstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationTexas Instruments TI83, TI83 Plus Graphics Calculator I.1 Systems of Linear Equations
Part I: Texas Instruments TI83, TI83 Plus Graphics Calculator I.1 Systems of Linear Equations I.1.1 Basics: Press the ON key to begin using your TI83 calculator. If you need to adjust the display contrast,
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationBasic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.
Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required
More informationThe Inverse of a Matrix
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
More informationSection 8.2 Solving a System of Equations Using Matrices (Guassian Elimination)
Section 8. Solving a System of Equations Using Matrices (Guassian Elimination) x + y + z = x y + 4z = x 4y + z = System of Equations x 4 y = 4 z A System in matrix form x A x = b b 4 4 Augmented Matrix
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationBrief Introduction to Vectors and Matrices
CHAPTER 1 Brief Introduction to Vectors and Matrices In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vectorvalued
More informationHelpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:
Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants
More informationTypical Linear Equation Set and Corresponding Matrices
EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationSection 10.4 Vectors
Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such
More informationWe know a formula for and some properties of the determinant. Now we see how the determinant can be used.
Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we
More informationReduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:
Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of EquationsGraphically and Algebraically Solving Systems  Substitution Method Solving Systems  Elimination Method Using Dimensional Graphs to Approximate
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More informationThe basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23
(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, highdimensional
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationThis assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the
More informationSYSTEMS OF EQUATIONS
SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More informationLecture notes on linear algebra
Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra
More information3. Solve the equation containing only one variable for that variable.
Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationElementary Matrices and The LU Factorization
lementary Matrices and The LU Factorization Definition: ny matrix obtained by performing a single elementary row operation (RO) on the identity (unit) matrix is called an elementary matrix. There are three
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationUsing row reduction to calculate the inverse and the determinant of a square matrix
Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible
More informationMathQuest: Linear Algebra. 1. Which of the following matrices does not have an inverse?
MathQuest: Linear Algebra Matrix Inverses 1. Which of the following matrices does not have an inverse? 1 2 (a) 3 4 2 2 (b) 4 4 1 (c) 3 4 (d) 2 (e) More than one of the above do not have inverses. (f) All
More informationSECTION 91 Matrices: Basic Operations
9 Matrices and Determinants In this chapter we discuss matrices in more detail. In the first three sections we define and study some algebraic operations on matrices, including addition, multiplication,
More information8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
More informationSystems of Linear Equations
Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11]. Main points in this section: 1. Definition of Linear
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More information1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient
Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear
More informationOverview. Essential Questions. Precalculus, Quarter 3, Unit 3.4 Arithmetic Operations With Matrices
Arithmetic Operations With Matrices Overview Number of instruction days: 6 8 (1 day = 53 minutes) Content to Be Learned Use matrices to represent and manipulate data. Perform arithmetic operations with
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationMath 1050 Khan Academy Extra Credit Algebra Assignment
Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for inclass presentation
More informationMBA Jump Start Program
MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationMath 240: Linear Systems and Rank of a Matrix
Math 240: Linear Systems and Rank of a Matrix Ryan Blair University of Pennsylvania Thursday January 20, 2011 Ryan Blair (U Penn) Math 240: Linear Systems and Rank of a Matrix Thursday January 20, 2011
More informationChapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints
Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationMath 313 Lecture #10 2.2: The Inverse of a Matrix
Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is
More informationMATH 551  APPLIED MATRIX THEORY
MATH 55  APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More informationLecture Notes: Matrix Inverse. 1 Inverse Definition
Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,
More informationVectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.
Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type
More informationSome Lecture Notes and InClass Examples for PreCalculus:
Some Lecture Notes and InClass Examples for PreCalculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
More informationSolving Systems of 3x3 Linear Equations  Elimination
Solving Systems of 3x3 Linear Equations  Elimination We will solve systems of 3x3 linear equations using the same strategies we have used before. That is, we will take something we don t recognize and
More informationElements of a graph. Click on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and yintercept in the equation of a line Comparing lines on
More information